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National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2017 Macedonia JBMO TST
2017 Macedonia JBMO TST
Part of
JBMO TST - Macedonia
Subcontests
(6)
Source
1
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Macedonia JBMO TST 2017
[url=https://artofproblemsolving.com/community/c675693]Macedonia JBMO TST 2017[url=http://artofproblemsolving.com/community/c6h1663908p10569198]Problem 1. Let
p
p
p
be a prime number such that
3
p
+
10
3p+10
3
p
+
10
is a sum of squares of six consecutive positive integers. Prove that
p
−
7
p-7
p
−
7
is divisible by
36
36
36
.[url=http://artofproblemsolving.com/community/c6h1663916p10569261]Problem 2. In the triangle
A
B
C
ABC
A
BC
, the medians
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
, and
C
C
1
CC_1
C
C
1
are concurrent at a point
T
T
T
such that
B
A
1
=
T
A
1
BA_1=TA_1
B
A
1
=
T
A
1
. The points
C
2
C_2
C
2
and
B
2
B_2
B
2
are chosen on the extensions of
C
C
1
CC_1
C
C
1
and
B
B
2
BB_2
B
B
2
, respectively, such that C_1C_2 = \frac{CC_1}{3} \text{and} B_1B_2 = \frac{BB_1}{3}. Show that
T
B
2
A
C
2
TB_2AC_2
T
B
2
A
C
2
is a rectangle.[url=http://artofproblemsolving.com/community/c6h1663918p10569305]Problem 3. Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals such that
x
y
z
=
1
xyz=1
x
yz
=
1
. Show that
x
2
+
y
2
+
z
x
2
+
2
+
y
2
+
z
2
+
x
y
2
+
2
+
z
2
+
x
2
+
y
z
2
+
2
≥
3.
\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.
x
2
+
2
x
2
+
y
2
+
z
+
y
2
+
2
y
2
+
z
2
+
x
+
z
2
+
2
z
2
+
x
2
+
y
≥
3.
When does equality happen?[url=http://artofproblemsolving.com/community/c6h1663920p10569326]Problem 4. In triangle
A
B
C
ABC
A
BC
, the points
X
X
X
and
Y
Y
Y
are chosen on the arc
B
C
BC
BC
of the circumscribed circle of
A
B
C
ABC
A
BC
that doesn't contain
A
A
A
so that
∡
B
A
X
=
∡
C
A
Y
\measuredangle BAX = \measuredangle CAY
∡
B
A
X
=
∡
C
A
Y
. Let
M
M
M
be the midpoint of the segment
A
X
AX
A
X
. Show that
B
M
+
C
M
>
A
Y
.
BM + CM > AY.
BM
+
CM
>
A
Y
.
[url=http://artofproblemsolving.com/community/c6h1663922p10569370]Problem 5. Find all the positive integers
n
n
n
so that
n
n
n
has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of
n
n
n
.
5
1
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Digits, prime factors, and exponents [Macedonia JBMO TST 2017, P5]
Find all the positive integers
n
n
n
so that
n
n
n
has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of
n
n
n
.
4
1
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Show that BM + CM > AY [Macedonia JBMO TST 2017, P4]
In triangle
A
B
C
ABC
A
BC
, the points
X
X
X
and
Y
Y
Y
are chosen on the arc
B
C
BC
BC
of the circumscribed circle of
A
B
C
ABC
A
BC
that doesn't contain
A
A
A
so that
∡
B
A
X
=
∡
C
A
Y
\measuredangle BAX = \measuredangle CAY
∡
B
A
X
=
∡
C
A
Y
. Let
M
M
M
be the midpoint of the segment
A
X
AX
A
X
. Show that
B
M
+
C
M
>
A
Y
.
BM + CM > AY.
BM
+
CM
>
A
Y
.
3
1
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cyc sum (x^2+y^2+z)/(x^2+2) \geq3 [Macedonia JBMO TST 2017, P3]
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive reals such that
x
y
z
=
1
xyz=1
x
yz
=
1
. Show that
x
2
+
y
2
+
z
x
2
+
2
+
y
2
+
z
2
+
x
y
2
+
2
+
z
2
+
x
2
+
y
z
2
+
2
≥
3.
\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.
x
2
+
2
x
2
+
y
2
+
z
+
y
2
+
2
y
2
+
z
2
+
x
+
z
2
+
2
z
2
+
x
2
+
y
≥
3.
When does equality happen?
2
1
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Rectangle and medians [Macedonia JBMO TST 2017]
In the triangle
A
B
C
ABC
A
BC
, the medians
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
, and
C
C
1
CC_1
C
C
1
are concurrent at a point
T
T
T
such that
B
A
1
=
T
A
1
BA_1=TA_1
B
A
1
=
T
A
1
. The points
C
2
C_2
C
2
and
B
2
B_2
B
2
are chosen on the extensions of
C
C
1
CC_1
C
C
1
and
B
B
2
BB_2
B
B
2
, respectively, such that C_1C_2 = \frac{CC_1}{3} \text{and} B_1B_2 = \frac{BB_1}{3}. Show that
T
B
2
A
C
2
TB_2AC_2
T
B
2
A
C
2
is a rectangle.
1
1
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3p+10 is a sum of six consecutive positive integers => 36 | p-7
Let
p
p
p
be a prime number such that
3
p
+
10
3p+10
3
p
+
10
is a sum of squares of six consecutive positive integers. Prove that
p
−
7
p-7
p
−
7
is divisible by
36
36
36
.